The effects of compressibility, hydrodynamic interaction and inertia on two-point, passive microrheology of viscoelastic materials

Abstract

In two-point passive microrheology, a modification of the original one-point technique the cross-correlations of two micron-sized beads embedded in a viscoelastic material are used to estimate the dynamic modulus of a material. The two-point technique allows for sampling of larger length scales which means that it can be used in materials with a coarser microstructure. An optimal separation between the beads exists at which the desired length and time scales are sampled while keeping an acceptable signal-to-noise-ratio in the cross-correlations. A larger separation can reduce the effect of higher-order reflections, but will increase the effects of medium inertia and reduce the signal-to-noise-ratio. The modeling formalisms commonly used to relate two-bead cross-correlations to the dynamic modulus and the complex Poisson ratio neglect inertia effects and underestimate the effect of reflections. A simple dimensional analysis suggests that for a model viscoelastic solid there exists a very narrow window of bead separation and frequency range where these effects can be neglected. In a recent work [Phys. Fluids, 2012, 24, 073103] we proposed an analysis formalism that accounts for medium inertia and high-order hydrodynamic reflections and therefore significantly increases the versatility of the two-point microrheology technique. In this paper we extend our analysis to compressible viscoelastic solids. There has been a recent interest in using two-point microrheology to measure the complex Poisson ratio of biopolymers [Das and MacKintosh, Phys. Rev. Lett., 2010, 105, 138102] however a rigorous analysis of the sensitivity of the technique to the static and dynamic properties of the Poisson ratio is still lacking. There are two decoupled statistics that can be followed with such a technique: motion parallel and perpendicular to the line of centers of the probe beads. We show that the cross-correlation in the direction parallel to the line of centers is insensitive to compressibility, so may reliably be used to determine G* (dynamic modulus) alone. Although, the cross-correlation in the perpendicular direction may then be used to extract a constant Poisson ratio, it is relatively insensitive to its frequency dependence. We consider the example of a composite actin/microtubule network.